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Theorem e2ebindALT 39688
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in e2ebindVD 39671. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
e2ebindALT (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))

Proof of Theorem e2ebindALT
StepHypRef Expression
1 axc11n 2462 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 nfe1 2183 . . . 4 𝑦𝑦𝜑
3219.9 2228 . . 3 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
4 excom 2198 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
5 nfa1 2184 . . . . . 6 𝑦𝑦 𝑦 = 𝑥
6 id 22 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)
7 biid 251 . . . . . . . . 9 (𝜑𝜑)
87a1i 11 . . . . . . . 8 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
98drex1 2477 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
106, 9syl 17 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
115, 10alrimi 2238 . . . . 5 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑))
12 exbi 1923 . . . . 5 (∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
1311, 12syl 17 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
14 bitr 800 . . . . . 6 (((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) ∧ (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
1514ex 397 . . . . 5 ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) → ((∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)))
1615impcom 394 . . . 4 (((∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑) ∧ (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑)) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
174, 13, 16sylancr 569 . . 3 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
18 bitr3 341 . . . 4 ((∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑) → ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑) → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑)))
1918impcom 394 . . 3 (((∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑) ∧ (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)) → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
203, 17, 19sylancr 569 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
211, 20syl 17 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
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